Probability Distributions

Transcription

1 STATGRAPHICS Re Probability Distributions Summary... Data Input... Analysis Summary... 3 Analysis Options... 3 Cumulatie Distribution... 4 Inerse CDF... 5 Random Numbers... 6 DensityMass Funtion... 9 CDF... 0 Surior Funtion... Log Surior Funtion... Hazard Funtion... Sae Results... Definitions... 3 Summary The Probability Distributions proedure performs arious operations for any of 46 probability distributions. In partiular, you may:. Plot the probability mass or density funtion, umulatie distribution, surior funtion, log surior funtion, or hazard funtion.. Calulate the umulatie distribution or inerse umulatie distribution. 3. Generate random numbers. Sample StatFolio: probdist.sgp Sample Data: None. 03 by StatPoint Tehnologies, In. Probability Distributions -

2 Data Input The data input dialog bo is used to selet the distribution to be ealuated. STATGRAPHICS Re Distribution: selet one of the 46 distributions listed. 03 by StatPoint Tehnologies, In. Probability Distributions -

3 STATGRAPHICS Re Analysis Summary The Analysis Summary shos the distribution seleted and the alues of its parameters. Probability Distributions Distribution: Normal Parameters: Mean Std. De. Dist. 0 Dist. 0 Dist Dist. 4 Dist. 5 Analysis Options Speify the up to 5 sets of parameters for the seleted distribution. The parameters required depend on the distribution seleted on the data input dialog bo. Definitions of the arious distributions are gien at the end of this doument. 03 by StatPoint Tehnologies, In. Probability Distributions - 3

4 STATGRAPHICS Re Cumulatie Distribution This pane shos the alue of the umulatie distribution and probability density or mass funtion at up to 5 alues of X. Cumulatie Distribution Distribution: Normal Loer Tail Area < Variable Dist. Dist. Dist. 3 Dist. 4 Dist Probability Density Variable Dist. Dist. Dist. 3 Dist. 4 Dist Upper Tail Area > Variable Dist. Dist. Dist. 3 Dist. 4 Dist Inluded in the table are: Loer Tail Area: the probability that a random ariable from the speified distribution is less than the alue shon in the leftmost olumn. Probability Density ontinuous distributions only: the height of the probability density funtion fx at the alue shon in the leftmost olumn. Probability Mass disrete distributions only: the probability that X equals the alue shon in the leftmost olumn. Upper Tail Area: the probability that a random ariable from the speified distribution is greater than the alue shon in the leftmost olumn. For eample, FX = at X=0.5 for the first distribution in the table aboe. Pane Options Random Variable: speify up to 5 alues at hih the umulatie distribution ill be alulated. 03 by StatPoint Tehnologies, In. Probability Distributions - 4

5 STATGRAPHICS Re Inerse CDF The Inerse CDF Cumulatie Distribution Funtion alulates the alue of the random ariable X at or belo hih lies a speified probability. Inerse CDF Distribution: Normal CDF Dist. Dist. Dist. 3 Dist. 4 Dist In the ase of a ontinuous distribution, the alue of X is alulated suh that the umulatie distribution funtion FX equals the probability shon in the leftmost olumn. In the ase of a disrete distribution, the alue displayed is the smallest alue of X suh that FX is greater than or equal to the probability shon in the leftmost olumn. For eample, FX = 0.0 at X= for the first distribution in the table aboe. Pane Options CDF: speify up to 5 alues of the umulatie distribution funtion at hih alues of X ill be determined. The alues must be greater than 0.0 and less than by StatPoint Tehnologies, In. Probability Distributions - 5

6 STATGRAPHICS Re Random Numbers Selet this pane to generate random numbers from the seleted distribution. The steps for generating random numbers are:. Speify the type of probability distribution on the data input dialog bo.. Use the Analysis Options dialog bo to speify the parameters of the distribution. 3. Selet Random Numbers from the list of tabular options and then selet Pane Options. 4. On the Pane Options dialog bo, speify ho many random numbers should be generated. 5. Selet Sae Results to plae the random numbers in the datasheet. Eah time you selet Sae Results, a different set of random numbers ill be generated. The method used for generating random numbers depends on the distribution seleted. Many of the distributions use the inerse transformation method, in hih a set of n random numbers U i are generated from a uniform distribution oer the interal 0, and then onerted to the desired distribution by letting X F i U i The uniform random numbers are generated using three linear ongruential generators in a manner designed to yield the same random sequene on any omputer gien the same seed. STATGRAPHICS sets the seed based on the time it is loaded, so that eah session ill generate a different sequene of random numbers. The methods for generating random numbers are summarized belo: Distribution Bernoulli Binomial Disrete uniform Geometri Hypergeometri Negatie binomial Poisson Beta Beta 4-parameter Method If Up, X=. Else X=0. Sum of n Bernoulli random ariables intuniforma,b+ ln U int i ln p Generation of m suesses from finite population of size n ithout replaement k + sum of k geometri random ariables Uses relationship beteen Poisson and eponential random ariables see La and Kelton. Y Y Y here Y ~gamma, and Y ~gamma, Translated beta random ariable. Birnbaum-Saunders Z 4Z 4 here Z ~ normal0, 03 by StatPoint Tehnologies, In. Probability Distributions - 6

7 STATGRAPHICS Re Cauhy Z Z here Z ~normal0, and Z ~normal0, Chi-square gamma,0.5 Erlang lnu i i Eponential Inerse transform method. Eponential - Translated eponential random ariable. parameter Eponential poer Numerial inerse transform method. F Y Y here Y ~hisquare and Y ~hisquare Folded normal X here X ~ normal, Gamma If =, eponential. Else aeptane-reetion method see La and Kelton. Gamma 3-parameter Translated gamma random ariable. Generalized gamma Numerial inerse transform method. Generalized logisti Numerial inerse transform method. Half-normal Generate X~normal,. If X, return X. Else return -X. Inerse Gaussian MiaelShuanyHass method see Gentle Laplae Inerse transform method. Largest etreme alue Inerse transform method. Logisti Inerse transform method. Loglogisti Inerse transform method. Loglogisti 3- Translated loglogisti random ariable. parameter Lognormal ep[normal,] Lognormal 3- Translated lognormal random ariable. parameter Maell a X X X 3 here X, X, and X 3 ~ normal0,b Nonentral hi-square If is integer, Nonentral F i Z i here Z i ~normal0,. Else numerial inerse transform method. Y Y 03 by StatPoint Tehnologies, In. Probability Distributions - 7

8 Nonentral t STATGRAPHICS Re here Y ~nonentral hisquare, and Y ~hisquare Z Y here Z ~normal0, and Y ~hisquare Normal Polar method see La and Kelton. Pareto Inerse transform method. Pareto -parameter Translated Pareto random ariable. Rayleigh a b logu Smallest etreme alue Student s t Inerse transform method. Z Y here Z ~normal0, and Y ~hisquare Triangular Inerse transform method. U Inerse transform method. Uniform a+b-au Weibull Inerse transform method. Weibull 3-parameter Translated Weibull random ariable. Pane Options Size: selet the number n of random numbers to be generated. After seleting the size, lik on Sae Results to sae the random numbers to the datasheet. 03 by StatPoint Tehnologies, In. Probability Distributions - 8

9 density STATGRAPHICS Re DensityMass Funtion This pane plots the probability density funtion fx for ontinuous distributions or the probability mass funtion p for disrete distributions. Normal Distribution Mean,Std. De. 0, 0, 0, For a ontinuous distribution suh as the normal distribution, the area under the density funtion oer an interal of alues for X equals the probability that X falls ithin that interal. When plotting the p.d.f. for a single, ontinuous distribution, Pane Options an be used to speify areas that ill be shaded on the plot: Shading: speifies one or more regions to be shaded. The speified areas ill be indiated on the plot and the probabilities assoiated ith the sum of all shaded areas ill be displayed: 03 by StatPoint Tehnologies, In. Probability Distributions - 9

10 umulatie probability density STATGRAPHICS Re Normal Distribution Probability = Mean,Std. De. 0, CDF This pane plots the umulatie distribution funtion FX. Normal Distribution Mean,Std. De. 0, 0, 0, FX equals the probability that the random ariable ill be less than or equal to X. 03 by StatPoint Tehnologies, In. Probability Distributions - 0

11 log surial prob. surial probability STATGRAPHICS Re Surior Funtion This pane plots the surior funtion SX, defined by SX = FX here FX is the umulatie distribution funtion. Normal Distribution Mean,Std. De. 0, 0, 0, SX equals the probability that the random ariable ill be greater than X. The name of the funtion is deried from situations here X represents an indiidual s or produt s lifetime. In that ase, SX is the probability that an indiidual suries at least X time units. Log Surior Funtion This pane plots the log of the surior funtion SX, hih is defined by SX = FX 3 Normal Distribution 3-7 Mean,Std. De. 0, 0, 0, by StatPoint Tehnologies, In. Probability Distributions -

12 hazard STATGRAPHICS Re Hazard Funtion The hazard funtion represents the onditional distribution of a random ariable gien than it is at least X. For ontinuous distributions, it is defined by HX = f SX 4 here f is the probability density funtion and SX is the surior funtion. For disrete distributions, it is defined by HX = p+ SX 5 here p is the probability mass funtion. Normal Distribution Mean,Std. De. 0, 0, 0, In life data analysis, the hazard funtion represents the onditional failure rate, i.e., the probability of failure in the net small inrement of time gien that an indiidual has suried until time X. Sae Results You an sae the folloing results to the datasheet: Random number for Dist. #: a set of random numbers generated from the speified distribution. The size of the set is determined from the Pane Options dialog bo for the Random Numbers pane. 03 by StatPoint Tehnologies, In. Probability Distributions -

13 STATGRAPHICS Re Definitions STATGRAPHICS generates results for 46 different probability distributions, 7 for disrete random ariables and the other 39 for ontinuous random ariables. Eah of the distributions ontains or more parameters, hih are either speified by the user or estimated from a data sample. Bernoulli Distribution Range of X: 0 or Common use: representation of an eent ith possible outomes. In the distributions belo, the primary outome ill be referred to as a suess. PMF: p p p Parameters: eent probability 0 p p p-p Binomial Distribution Range of X: 0,,,, n Common use: distribution of number of suesses in a sample of n independent Bernoulli trials. Commonly used for number of defets in a sample of size n. PMF: p n n p p Parameters: eent probability 0 p, number of trials n. np np-p Disrete Uniform Distribution Range of X: a, a+, a+,, b Common use: distribution of an integer alued ariable ith both a loer bound and an upper bound. PMF: p b a Parameters: loer limit a, upper limit b a. a b b a Geometri Distribution Range of X: 0,,, Common use: aiting time until the ourrene of the first suess in a sequene of independent Bernoulli trials. Number of items inspeted before the first defet is found. PMF: p p p Parameters: eent probability 0 p p p p p 03 by StatPoint Tehnologies, In. Probability Distributions - 3

14 Hypergeometri Distribution Range of X: ma0,n-m,,,, minm,n STATGRAPHICS Re Common use: number of items of a gien type seleted from a finite population ith to types of items, suh as good and bad. Aeptane sampling from lots of fied size. PMF: m N m n p N n Parameters: population size N, number of items 0 m N, sample size n mn N mn m N n N N N Negatie Binomial Pasal Distribution Range of X: 0,,, Common use: aiting time until the ourrene of k suesses in a sequene of independent Bernoulli trials. Number of good items inspeted before the k th defet is found. PMF: k k p p p Parameters: eent probability p, number of suesses k p k p k p p NOTE: the definition of this distribution has hanged from earlier ersions. Earlier ersions inluded k as part of the definition of the random ariable, so that the range of X as k or greater instead of 0 or greater. The hange has been made to allo the negatie binomial distribution to be more easily used as a model for oerdispersed ount data, i.e, integer data in hih the ariane eeeds the mean. Poisson Distribution Range of X: 0,,, Common use: number of eents ourred in an interal of fied size hen eents our independently. Common model for number of defets per unit. PMF: e p! Parameters: mean > 0 Beta Distribution Range of X: 0 X Common use: distribution of a random proportion. f B, Parameters: shape > 0, shape > 0 03 by StatPoint Tehnologies, In. Probability Distributions - 4

15 STATGRAPHICS Re Beta Distribution 4-parameter Range of X: a X b Common use: model for ariable ith both loer and upper limits. Often used as a prior distribution for Bayesian analysis. f a b B, b a Parameter: shape > 0, shape > 0, loer limit a, upper limit b > a a b b a Birnbaum-Saunders Distribution Range of X: X > 0 Common use: model for the number of yles needed to ause a rak to gro to a size that ould ause a frature to our. f here z is the standard normal pdf Parameters: shape > 0, sale > Cauhy Distribution Range of X: all real X Common use: model for measurement data ith longer and flatter tails than the normal distribution. f Parameters: mode, sale > 0 not defined not defined Chi-Squared Distribution Range of X: X 0 Common use: distribution of the sample ariane s from a normal population. 03 by StatPoint Tehnologies, In. Probability Distributions - 5

16 f e Parameters: degrees of freedom STATGRAPHICS Re Erlang Distribution Range of X: X 0 Common use: length of time before arrials in a Poisson proess. f e Parameters: integer shape, sale > 0 Eponential Distribution Range of X: X 0 Common use: time beteen onseutie arrials in a Poisson proess. Lifetime of items ith a onstant hazard rate. f e Parameters: mean > 0 Eponential Distribution -parameter Range of X: X Common use: model for lifetimes ith a loer limit. f e Parameters: threshold, sale > 0 Eponential Poer Distribution Range of X: all real X Common use: symmetri distribution ith parameter ontrolling the kurtosis. Speial ases inlude normal and Laplae distributions. 03 by StatPoint Tehnologies, In. Probability Distributions - 6

17 STATGRAPHICS Re by StatPoint Tehnologies, In. Probability Distributions - 7 ep f Parameters: mean, shape -, sale > 0 3 F Distribution Range of X: X 0 Common use: distribution of the ratio of to independent ariane estimates from a normal population. f Parameters: numerator degrees of freedom > 0, denominator degrees of freedom > 0 if > 4 if > 4 Folded Normal Distribution Range of X: X 0 Common use: absolute alues of data that follos a normal distribution. osh e f Parameters: loation > 0, sale 0 ep here z is the standard normal df ep erf Gamma Distribution Range of X: X 0 Common use: model for positiely skeed measurements. Time to omplete a task, suh as a repair. e f Parameters: shape >0, sale > 0

18 STATGRAPHICS Re Gamma Distribution 3-parameter Range of X: X Common use: model for positiely skeed data ith a fied loer bound. e f Parameters: shape >0, sale > 0, threshold Generalized Gamma Distribution Range of X: X > 0 Common use: general distribution ontaining the eponential, gamma, Weibull, and lognormal distributions as speial ases. ln f ep ep ln here = [log ]. Parameters: loation, sale > 0, shape > 0 ep ln ep ln Generalized Logisti Distribution Range of X: all real Common use: used for the analysis of etreme alues. May be either left-skeed or right-skeed, depending on the shape parameter. f ep ep Parameters: loation, sale > 0, shape > here z is the digamma funtion ' 6 03 by StatPoint Tehnologies, In. Probability Distributions - 8

19 Half Normal Distribution Range of X: X Common use: normal distribution folded about its mean. f ep Parameters: sale > 0, threshold STATGRAPHICS Re Inerse Gaussian Distribution Range of X: X > 0 Common use: first passage time in Bronian motion. f ep z z e e z Parameters: mean > 0, sale > 0 here z = ln Laplae Double Eponential Distribution Range of X: all real X Common use: symmetri distribution ith a ery pronouned peak and long tails f e Parameters: mean, sale > 0 Largest Etreme Value Distribution Range of X: all real X Common use: distribution of the largest alue in a sample from many distributions. Also used for positiely skeed measurement data. f ep ep Parameters: 6 Logisti Distribution Range of X: all real X 03 by StatPoint Tehnologies, In. Probability Distributions - 9

20 Common use: used as a model for groth and as an alternatie to the normal distribution. f ep z ep z here Parameters: mean, standard deiation z STATGRAPHICS Re Loglogisti Distribution Range of X: X > 0 Common use: used for data here the logarithms follo a logisti distribution. f ep z ep z Parameters: median ep, shape ep here ln z ep Loglogisti Distribution 3-parameter Range of X: X > Common use: used for data here the logarithms follo a logisti distribution after subtrating a threshold alue. f ep z ep z here Parameters: median ep, sale threshold ep ln z ep Lognormal Distribution Range of X: X > 0 Common use: used for data here the logarithms follo a normal distribution. f ln e Parameters: loation, sale > 0 e e e Lognormal Distribution 3-parameter Range of X: X > Common use: used for data here the logarithms follo a normal distribution after subtrating a threshold alue. f ln e Parameters: loation, sale > 0, threshold e e e 03 by StatPoint Tehnologies, In. Probability Distributions - 0

21 STATGRAPHICS Re by StatPoint Tehnologies, In. Probability Distributions - Maell Distribution Range of X: X > Common use: the speed of a moleule in an ideal gas. 3 ep f Parameters: sale, threshold Nonentral Chi-squared Distribution Range of X: X 0 Common use: used to alulate the poer of hi-squared tests. e e f! 0 Parameters: degrees of freedom > 0, nonentrality 0 Nonentral F Distribution Range of X: X 0 Common use: used to alulate the poer of F tests. 0,! B e f Parameters: number degrees of freedom > 0, denominator degrees of freedom > 0, nonentrality > 0 if > 4 if > 4 Nonentral t Distribution Range of X: all real X Common use: used to alulate the poer of t tests. 0! e f Parameters: degrees of freedom > 0, nonentrality 0 Normal Distribution

22 Range of X: all real X Common use: idely used for measurement data, partiularly hen ariability is due to many soures. f e Parameters: mean, standard deiation > 0 STATGRAPHICS Re Pareto Distribution Range of X:X Common use: model for many soio-eonomi quantities ith ery long upper tails. f Parameters: shape > 0 if > if > Pareto Distribution -parameter Range of X:X Common use: distribution of soio-eonomi quantities ith a loer bound. f Parameters: shape >0, threshold 0 if > if > Rayleigh Distribution Range of X:X > Common use: the distane beteen neighboring items in a pattern generated by a Poisson proess. f Parameters: sale, threshold 4 ep Smallest Etreme Value Distribution Range of X: all real X Common use: distribution of the smallest alue in a sample from many distributions. Also used for negatiely skeed measurement data. f ep ep 03 by StatPoint Tehnologies, In. Probability Distributions -

23 Parameters: mode, sale > 0 6 STATGRAPHICS Re Student s t Distribution Range of X: all real X Common use: referene distribution for the sample mean hen sampling from a normal population ith unknon ariane. f Parameters: degrees of freedom 0 if > Triangular Distribution Range of X: a X b Common use: often used as a rough model in the absene of data. f f a, ba a b, ba b Parameters: loer limit a, mode a, upper limit b a b 3 a b abab 8 U Distribution Range of X: b - a X b + a Common use: used in metrology for the distribution of quantities that osillate around a speifi alue. f a b a Parameters: sale a > 0, mean b b a Uniform Distribution Range of X: a X b Common use: model for ariable ith equal probability eeryhere oer an interal. 03 by StatPoint Tehnologies, In. Probability Distributions - 3

24 f b a Parameters: loer limit a, upper limit ba a b b a STATGRAPHICS Re Weibull Distribution Range of X: X 0 Common use: idely used in reliability analysis to model produt lifetimes. f e Parameters: shape > 0, sale > 0 Weibull Distribution 3-parameter Range of X: X Common use: used for produt lifetimes ith a loer bound. f ep Parameters: shape > 0, sale > 0, threshold 03 by StatPoint Tehnologies, In. Probability Distributions - 4